From: Clark <clark@brutele.be>
Subject: Re: Foundational matters Was: Re: Axiom of Choice
Date: Wed, 12 Jul 2000 12:41:40 +0200
Newsgroups: sci.math
To: Stephen Montgomery-Smith <stephen@math.missouri.edu>
Summary: [missing]

Stephen Montgomery-Smith wrote:
> 
> Clark wrote:
> >
> > Coming at this from a slightly different angle, there's a (?) well-known
> > justification (2nd order) of Peano's axioms (including induction) by
> > Frege from Hume's Principle (number of F's = number of G's iff F's can
> > be one-one correlated with G's). (Hume's Principle doesn't suffer the
> > fate of Frege's axiom V, as various people - Boolos, Burgess, Hodes ...
> > - have noted.)
> 
> I would be interested to know this argument.  If it is short, could
> you share it on the internet?

The essence of it is short enough, I think. (I'll give references for
details below.) 

Hume's Principle: 
Nx:Fx=Nx:Gx iff (ER)(Fx1-1(R)Gx) 
['N'say'number of' ...'1-1(R)'say'1-1 related by R to' (standardly
defined)] 

First, use Hume's Principle to get cardinality by abstraction: 
(F)(Ey)(y=Nx:Fx)

Next, define 
zero [0=Nx:x!=x] and 
x Precedes y [Pxy iff (EF)(Ez)(Fz&y=Nw:Fw&x=Nv:(Fv&v!=z))]
[Prove that P is 1-1]

Now define the ancestral relation R* of a relation R:
R*xy iff (F)[((z)(Rxz->Fz)&(v)(w)((Fv&Rvw)->Fw))->Fy]
[Prove that Rxy->R*xy and that R* is transitive]

So we can define natural number:
Natx iff (0=x or P*0x)

And now it's easy to see that induction
(F)[F(0)&(x)(Fx->(y)(Pxy->F(y)))->(x)(Natx->Fx)]
follows
[take each conjunct of the antecedent of P*0x separately ... deduce Fx]


This might seem a bit of a damp squib at first. Hasn't Frege just
defined natural numbers as things that induction works on by means of
the more general notion of ancestral? There may be something in this.
But, on the other hand, doing the work this way gives us a view of the
Peano axioms as a whole (the other axioms can also be derived as logical
consequences of Natx), and as consequences just of the notion of
cardinals with ancestral precedence.

Now it looks as though the only non-logical notion in play here is just
that of Hume's Principle. Is it true ... if so, is it more or less
obvious than the principle of induction?

Frege, of course, tried - and, as Russell showed him, failed - to derive
Hume's Principle logically.

[References: Much of the above can be found at slightly greater length
in Crispin Wright's 'Frege's Conception of Numbers as Objects', (chapter
4) which pretty much set the whole neo-Fregean ball rolling, I think.
Many relevant subsequent papers are collected in William Demopoulos
(ed.) 'Frege's Philosophy of Mathematics'.]

I'm much more of a neophyte than an expert with all this, by the way -
and I'd appreciate criticism by those who are more expert, of which I'm
sure there are many around.

Bob